Hypercomplex Widely Linear Estimation Through the Lens of Underpinning Geometry
Hypercomplex Widely Linear Estimation Through the Lens of Underpinning Geometry
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DOI：
10.1109/tsp.2019.2922151
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发表时间：
2019-08-01
影响因子：
5.4
通讯作者：
Mandic, Danilo P.
中科院分区：
文献类型：
--
作者：
Nitta, Tohru;Kobayashi, Masaki;Mandic, Danilo P.
我们提供了一个严格的帐户之间的等价性复值广泛的线性估计方法和四元数对合广泛的线性估计方法与向量值真实的线性估计。这是通过自由度的帐户，并通过提供一个复杂的变量和同构的二元真实的向量之间的矩阵映射，和一个四元数变量与一个四元真实的向量。此外，我们表明，在复值线性估计方法，复值广泛的线性估计方法，四元数线性估计方法，四元数半广泛的线性估计方法，和四元数对合广泛的线性估计方法中的参数包括不同的几何结构施加在复数和四元数，分别，而实值线性估计方法不表现出任何结构。这个关键的区别解释了，无论是在理论上还是在实践中，除法代数（复数，四元数）中的估计优于多元真实的向量。此外，我们还讨论了超复宽线性估计方法的估计量的计算复杂性。
We provide a rigorous account of the equivalence between the complex-valued widely linear estimation method and the quaternion involution widely linear estimation method with their vector-valued real linear estimation counterparts. This is achieved by an account of degrees of freedom and by providing matrix mappings between a complex variable and an isomorphic bivariate real vector, and a quaternion variable versus a quadri-variate real vector. Furthermore, we show that the parameters in the complex-valued linear estimation method, the complex-valued widely linear estimation method, the quaternion linear estimation method, the quaternion semi-widely linear estimation method, and the quaternion involution widely linear estimation method include distinct geometric structures imposed on complex numbers and quaternions, respectively, whereas the real-valued linear estimation methods do not exhibit any structure. This key difference explains, both in theoretical and practical terms, the advantage of estimation in division algebras (complex, quaternion) over their multivariate real vector counterparts. In addition, we discuss the computational complexities of the estimators of the hypercomplex widely linear estimation methods.